So far we have assumed that each individual i knows with certainty his valuation of each of the s services vis(mis), and, hence, given some q, the dollar amount of the different services that will be provided to him upon joining the plan. In order to make our model more realistic and to prepare for empirical application, we shall now allow for each individual to be uncertain about his future demands for the different services. Let us suppose that each individual has a set of prior beliefs about his possible health care demands, and that the plan shares these beliefs review.

Let T denote the set of possible health states of each individual and let t denote an element of T. Let vt ={vtl(mtl), vt2(mt2),…, vts(mts)} denote the vector of S valuation functions for the S services, if the health state is realized to be t. We assume that for each t and s, vts( ) satisfies the properties discussed earlier.

Upon joining the plan, each individual i is uncertain about his health state t, and he has some prior distribution fj over the set of possible states. The individual’s prior distribution fj represents what the individual believes about his health states. Momentarily we will focus on one individual and hence will drop the subscript i from the notation.

Let xt be some random variable, the value of which depends on the state t, and let f be a distribution function defined over T. Let Ef [xt] denote the expected value of xt with respect to the distribution f.

The order of moves in this modified model is as follows: first, the plan chooses its level of shadow prices q-(qi, q2>–*> 4s)> then the individual chooses whether or not to join the plan (in a manner studied below), and finally the individual’s health state is realized and services are provided.

We assume that when services are provided, the individual’s “true” health state is already known. Hence, for a given shadow price qs and a valuation function vts, the plan’s expenditures on this individual on services s will be mts(qs), given by:
If the individual joins the plan, his expected utility is: ц + Ef [vt(q)J. Note that unlike his health state, we assume that ц is known to the individual when choosing the plan.

Let ut denote the individual’s utility if his health state is t and he chooses the alternative plan. Thus, Ef [ut ] is the individual’s expected utility if he chooses the alternative plan.

We assume no asymmetry of information between the plan and the individual regarding the individual’s health state. Thus, the plan knows the individual’s prior beliefs, f, about his future health state. The plan, however, does not know the true value of \i and it holds beliefs Ф(ц) about its cumulative distribution.

Thus, for a given shadow price q, the plan’s assigned probability that the individual will join the plan if his prior beliefs about his future health state are given by fj is:
where mis = Efi[mts]is individual i’s predicted expenditures on services s, where the prediction is with respect to the individual’s prior beliefs about his future expenditures on service s. Define